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<b>let </b><font color="Maroon">GS</font> be   <a href="lexbfs.html#NM10">VLabelNumberingSeq</a>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>set </b><font color="Maroon">CSN</font> = <font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">VLN</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_1"><b>suppose </b></a><a NAME="E1:120_1_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:120_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:120_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_2"><b>suppose </b></a><a NAME="E1:120_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:120_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:120_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:120_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><a NAME="E2:120_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:120_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:120_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E5:120_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E6:120_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:120"/>
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">GS</font> <a href="glib_000.html#K40">.Lifespan()</a>  &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  implies 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> ) )
 <b>by </b><i/>;<br/>


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